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We give an overview of some questions and results in integral $p$-adic Hodge theory. A few proofs are supplied.

We describe a relation between two invariants that measure the complexity of a hypersurface singularity. One is the Hodge spectrum which is related to the monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The other is the multiplier ideal, having to do with log resolutions., Comment: shorter final version to appear in Math. Ann

We continue our work on variations of graded-polarized mixed Hodge structures by defining analogs of the harmonic metric equations for filtered bundles and proving a precise analog of Schmid's Nilpotent Orbit Theorem for 1-parameter degenerations of graded-polarized mixed Hodge structure., Correction to Theorem 5.11

To a Hodge structure V of weight k with CM by a field K we associate Hodge structures V_{-n/2} of weight k+n for n positive and, under certain circumstances, also for n negative. We show that these `half twists' come up naturally in the Kuga-Satake varieties of weight two Hodge structures with CM by an imaginary quadratic field., Comment: 16 pages. To appear in: Journal of the Mathematical Society of Japan

In the warped compactification with a single Randall-Sundrum brane, a puzzling claim has been made that scalar fields can be bound to the brane but their Hodge dual higher-rank anti-symmetric tensors cannot. By explicitly requiring the Hodge duality, a prescription to resolve this puzzle was recently proposed by Duff and Liu. In this note, we implement the Hodge duality via path integral formulation in the presence of the background gravity fields of warped compactifications. It is shown that the prescription of Duff and Liu can be naturally understood within this framework., 7 pages, LaTeX

In abstract Yang-Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible ("pseudoinstanton"). We suggest a metric-affine action which is a natural generalization of the Yang-Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino., Comment: 24 pages, LaTeX2e

Throughout this chapter X will denote a nonsingular projective surface defined over an algebraically closed field k of arbitrary characteristic, and K will denote a canonical divisor on X.

We study *-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz' first calculus, the calculation of the eigenvalues of the Laplacian, Comment: 39 pages, LaTeX2e

In the last 35 years or so a lot of effort was devoted to and great success achieved in the study of the singularities of an analytic function germ f : (ℂ n+1,0) ’ (ℂ, 0). Indeed, when f defines an isolated hypersurface singularity (IHS for short in the sequel) the topology of the situation was studied by Milnor [36] and Brieskorn [9] who have obtained fascinating relations to the exotic differentiable structures on spheres. Arnold and his school have classified the simplest IHS and brought into light unexpected relations to the classification of simple Lie algebras, du Val rational double points of surfaces and other classical objects in algebraic geometry, see for a complete presentation the monograph [2].

We extend the nonlinear Hodge decomposition of Iwaniec et al. [1] to other vector fields. We give applications to Leray-Lions type equations with very weak assumptions on data similar to those developed in [1].

Let G=G(t,z) be one of the N^2-dimensional bicovariant first order differential calculi for the quantum groups GL_q(N), SL_q(N), O_q(N), or Sp_q(N), where q is a transcendental complex number and z is a regular parameter. It is shown that the de Rham cohomology of Woronowicz' external algebra G^ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (twosided) coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) the cohomology ring is isomorphic to the coinvariant external algebra G^_{inv} and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. Keywords: quantum groups, bicovariant differential calculi, de Rham cohomology, Laplace-Beltrami operator, Hodge theory, Comment: LaTeX2e, 40 pages

Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge structures and we give a detailed account of the construction of KugaSatake varieties. The Hodge conjecture is discussed in section2. An excellent survey of the Hodge conjecture for abelian varieties is [G].

Let SU_X(n,L) be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g>1 curve X. Let SU_X^s(n,L) denote the open subset parameterizing stable bundles. We show that for small i, the mixed Hodge structure on H^i(SU_X^s(n, L), Q) is independent of the degree of L, and hence pure of weight i. Moreover any simple factors is, up to Tate twisting, isomorphic to a summand of a tensor power of H^1(X,Q). A more precise statement for i = 3, yields a Torelli theorem complementing earlier work of several authors. This is a replacement of our preprint Intermediate Jacobians of Moduli spaces which contained a gap., It was brought to our attention, by H. Esnault, that the hyperplane H in our thm 6.1.1 needs to be general. Further comments are contained in the text

We introduce a doubled formalism for the bosonic sector of the maximal supergravities, in which a Hodge dual potential is introduced for each bosonic field (except for the metric). The equations of motion can then be formulated as a twisted self-duality condition on the total field strength \G, which takes its values in a Lie superalgebra. This doubling is invariant under dualisations; it allows a unification of the gauge symmetries of all degrees, including the usual U-dualities that have degree zero. These ``superdualities'' encompass the dualities for all choices of polarisation (i.e. the choices between fields and their duals). All gauge symmetries appear as subgroups of finite-dimensional supergroups, with Grassmann coefficients in the differential algebra of the spacetime manifold., Latex, 58 pages, minor corrections

Throughout this paper M will be a compact Finsler space with positive definite metric. That is to say, M will be a boundaryless compact manifold equipped with a positive smooth ℝ-valued function F defined on the slit tangent bundle TM (the tangent bundle TM with the zero section removed) such that F(x, y) is a positive homogeneous function of degree one in y ∈ T x M for any x∈ M and the Finsler metric tensor defined by is positive definite. The latter is a tensor in the Finslerian sense, that is, such that the diagram commutes. (Here π1 and π2 are the projections of the bundles TM and T 2 M.)

One of the oldest and most important techniques in modern Riemannian geometry is that of the Bochner technique. In this chapter we shall prove some of the classical theorems Bochner proved about obstructions to the existence of Killing fields and harmonic 1-forms. We also explain how the Bochner technique extends to forms. This will in the next chapter lead us to a classification of compact manifolds with nonnegative curvature operator. To establish the relevant Bochner formula for forms, we have used the language of Clifford multiplication. It is, in our opinion, much easier to work consistently with Clifford multiplication, rather than trying to keep track of wedge products, interior products, Hodge star operators, exterior derivatives, and their dual counterparts. In addition, it has the effect of preparing one for the world of spinors, although we won’t go into this here. In the last section we give a totally different application of the Bochner technique. In effect, we try to apply it to the curvature tensor itself. The outcome will be used in the next chapter, where manifolds with nonnegative curvature operator will be classified. The Bochner technique on spinors is only briefly mentioned in this chapter, but Appendix C is devoted to this subject.

Hirzebruch’s construction of the signature complex, [H] (or [N], for an easy to read exposition), allowed for a proof of the Lichnerowicz Vanishing Theorem on a Riemannian manifold which may not be spin. The construction uses the Hodge star to define self-dual and anti-self-dual forms, for which the Hodge-deRham operator serves as a first order elliptic operator with the signature as its index. Then a traditional Weitzenbock formula for the Levi-Civita connection allows one to show a relationship between the scalar curvature and the (vanishing of) the signature of the manifold in question. We wish to generalize this construction to the case of Finsler spaces.

Recently Kaniel & Itin proposed a gravitational model with the wave type equation [\square+\lambda(x)]\vartheta^\alpha=0 as vacuum field equation, where \vartheta^\alpha denotes the coframe of spacetime. They found that the viable Yilmaz-Rosen metric is an exact solution of the tracefree part of their field equation. This model belongs to the teleparallelism class of gravitational gauge theories. Of decisive importance for the evaluation of the Kaniel-Itin model is the question whether the variation of the coframe commutes with the Hodge star. We find a master formula for this commutator and rectify some corresponding mistakes in the literature. Then we turn to a detailed discussion of the Kaniel-Itin model., Comment: 33 pages LATEX, one postscript figure included

Given a polynomial map $f:\Bbb C^{n+1}\to\Bbb C$, one can attach to it a geometrical variation of mixed Hodge structures (MHS) which gives rise to a limit MHS. The equivariant Hodge numbers of this MHS are analytical invariants of the polynomial map and reflect its asymptotic behaviour. In this paper we compute them for a class of generic polynomials, in terms of Hodge numbers attached to isolated hypersurface singularities and Hodge numbers of cyclic coverings of projective space branched along a hypersurface., AMS-LaTeX, 33 pages

Throughout this chapter F denotes a transversally oriented Riemannian foliation on a closed oriented manifold M. We discuss Hodge theory and a duality theorem for the cohomology of basic forms [K-To 10,12].

We owe Riemann the first steps in algebraic topology of manifolds. The modest germs of Riemann ideas had grown, as a result of works of Klein, Poincare, Brouwer, Lefschetz, Hopf, de Rham, Hodge, Kodaira, Leray, Serre, to mention only the names of the greatest, into big tree of the theory of (co)homology theory of Riemann manifolds. In this chapter, I will try to present the facts which are most spectacular and, at the same time, most clearly related to geometrical and analytical ideas of Riemann. These results are forever associated with the names of de Rham and Hodge. We owe Riemann a fundamental notion of the harmonic tensor field ω on M 2 and its period on a cycle c, that is, a close curve c on M 2: this is the integral ∫ c ω. We also owe Riemann the theorem on existence of a harmonic field of given periods on M 2.

Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of f. The construction is reminiscent of techniques from the theory of hyperplane arrangements, and relies on Arnold's calculation of the cohomology of the configuration space of the complex line. This resolution is S_n-equivariant. We apply it to the universal elliptic curve with complete level structure of level N>=3 over the modular curve Y(N), obtaining a formula for the S_n-equivariant Serre polynomial (Euler characteristic of H^*_c(V,Q) in the Grothendieck group of the category of mixed Hodge structures) of the moduli space M_{1,n}. In a sequel to this paper, this is applied in the calculation of the S_n-equivariant Hodge polynomial of the compactication \bar{M}_{1,n}., 25 pages. amslatex-1.2, pb-diagram and lamsarrow. There are a number of corrections from the first version

There is a classical relation between the p-adic absolute value of the eigenvalues of Frobenius on crystalline cohomology and Hodge numbers, for a variety in characteristic p: “the Newton polygon lies on or above the Hodge polygon” [14], [1]. For a variety in characteristic p with a lift to characteristic 0, Fontaine conjectured and Faltings proved a more precise statement: there is an inequality which relates the slope of Frobenius on any Frobenius-invariant subspace of the crystalline cohomology to the Hodge filtration, restricted to that subspace [7], [4]. A vector space over a p-adic field together with a σ-linear endomorphism and a filtration which satisfies this inequality is called a weakly admissible filtered isocrystal (see section 1 for the precise definition). The category of such objects is one possible p-adic analogue of the category of Hodge structures: in particular, it is an abelian category. We give a new proof of Faltings’s theorem that the tensor product of weakly admissible filtered isocrystals over a p-adic field is weakly admissible [5]. By a similar argument, we also prove a characterization of weakly admissible filtered isocrystals with G-structure in terms of geometric invariant theory, which was conjectured by Rapoport and Zink [19]. Before Faltings, Laffaille [12] had proved the tensor product theorem in the case of filtered isocrystals over an unramified extension of Qp. Faltings’s proof works by reducing this problem of σ-linear algebra to a different problem of pure linear algebra, the problem of showing that the tensor product of two vector spaces, each equipped with a finite “semistable” set of filtrations, is semistable. The latter problem is solved by constructing suitable integral lattices (in [5]) or hermitian metrics (in [20]) on vector spaces with a semistable set of filtrations, just as one can prove that the tensor product of semistable bundles on an algebraic curve is semistable using Narasimhan-Seshadri’s hermitian metrics ([6], [16]). In this paper, we can avoid the reduction from filtered isocrystals to filtered vector spaces. The point is that Ramanan and Ramanathan’s algebraic proof [17] that the tensor product of semistable vector bundles is semistable can be modified to apply directly to filtered isocrystals. We have an inequality to prove for a class of linear subspaces S of a tensor product V ⊗W . The inequality is obvious for sufficiently general subspaces S and also if S is a very special subspace, say if S is a decomposable subspace S1 ⊗ S2 ⊂ V ⊗W . But it is not clear how to prove the inequality we want if S is somewhere in the middle. The solution, following Ramanan and Ramanathan, is to use geometric invariant theory to give a sharp dichotomy between “general” subspaces and “special” subspaces of V ⊗W , in such a way that we get useful information in either case.

We consider a local variation of Hodge structure with abelian monodromy group. We show the existence of a canonical limit mixed Hodge structure, and we prove the nilpotent orbit theorem in this context.

The classic question of a nonabelian Yang-Mills analogue to electromagnetic duality is here examined in a minimalist fashion at the strictly 4-dimensional, classical field and point charge level. A generalisation of the abelian Hodge star duality is found which, though not yet known to give dual symmetry, reproduces analogues to many dual properties of the abelian theory. For example, there is a dual potential, but it is a 2-indexed tensor $T_{\mu\nu}$ of the Freedman-Townsend type. Though not itself functioning as such, $T_{\mu\nu}$ gives rise to a dual parallel transport, $\tilde{A}_\mu$, for the phase of the wave function of the colour magnetic charge, this last being a monopole of the Yang-Mills field but a source of the dual field. The standard colour (electric) charge itself is found to be a monopole of $\tilde{A}_\mu$. At the same time, the gauge symmetry is found doubled from say $SU(N)$ to $SU(N) \times SU(N)$. A novel feature is that all equations of motion, including the standard Yang-Mills and Wong equations, are here derived from a `universal' principle, namely the Wu-Yang (1976) criterion for monopoles, where interactions arise purely as a consequence of the topological definition of the monopole charge. The technique used is the loop space formulation of Polyakov (1980)., Comment: We regret that, due to a technical hitch, parts of the reference list were mixed up. This is the corrected version. We apologize to the authors whose papers were misquoted

The fact that the Lie group SO(4) is nonsimple gives 4-dimensional geometry an extremely distinctive flavor. Indeed, the choice of a Riemannian metric g on an oriented 4-manifold M splits the bundle of 2-forms $${\Lambda ^2} = {\Lambda ^ + } \oplus {\Lambda ^ - }$$ (1) into the rank-3 bundles of self-dual and anti-self dual 2-forms, respectively defined as the ±1-eigenspaces of the Hodge star operator ⋆ : ⋀2 → ⋀2; this just reflects the fact that the adjoint representation of SO(4) on the skew (4 x 4)-matrices is the sum of two 3-dimensional representations, as indicated by the Lie algebra isomorphism so(4) ≅ so(3)⊕so(3). The decomposition (1) is conformally invariant, in the sense that it is unchanged if g is replaced by ug for any positive function u; but reversing the orientation of M interchanges the bundles ⋀±.

We consider a total action composed of two Dirac–Born–Infeld (DBI) actions: one for a vector field Aμ and another for an axial vector field Bμ. We impose a duality condition [Formula: see text], where [Formula: see text] is the Hodge dual of Gμν, and g is a DBI interaction constant. Interestingly, there are two different global duality rotation symmetries in the presence of DBI interactions: (i) [Formula: see text], [Formula: see text], and (ii) δζAμ = - ζBμ, δζBμ = + ζAμ. Both of these symmetry are on-shell symmetries, including nonlinear higher-order terms. The remarkable aspect is that these symmetries are valid even in the presence of DBI interactions. The coupling of this system to N = 1 supergravity is also discussed.

Let $f: X \rightarrow S$ be a family of non singular projective varieties parametrized by a complex algebraic variety $S$. Fix $s \in S$, an integer $p$, and a class $h \in {\rm H}^{2p}(X_s,\Z)$ of Hodge type $(p,p)$. We show that the locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This result, which in the geometric case would follow from the rational Hodge conjecture, is obtained in the setting of variations of Hodge structures., Comment: 25 pages, Plain TeX

In order to account for all possible contractions allowed by the presence of the solder form, a generalized Hodge dual is defined for the case of soldered bundles. Although for curvature the generalized dual coincides with the usual one, for torsion it gives a completely new dual definition. Starting from the standard form of a gauge lagrangian for the translation group, the generalized Hodge dual yields precisely the lagrangian of the teleparallel equivalent of general relativity, and consequently also the Einstein-Hilbert lagrangian of general relativity., Comment: 8 pages, no figures. Accepted for publication in Journal of Physics A

This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be written in terms of a material Hodge star operator. In the discrete theory presented in this paper, instead of referring to continuum quantities, we postulate the existence of some discrete scalar-valued and vector-valued primal and dual differential forms on a discretized solid, which is assumed to be a triangulated domain. We find the discrete governing equations by requiring energy balance invariance under time-dependent rigid translations and rotations of the ambient space. There are several subtle differences between the discrete and continuous theories. For example, power of tractions in the discrete theory is written on a layer of cells with a nonzero volume. We obtain the compatibility equations of this discrete theory using tools from algebraic topology. We study a discrete Cosserat medium and obtain its governing equations. Finally, we study the geometric structure of linearized elasticity and write its governing equations in a matrix form. We show that, in addition to constitutive equations, balance of angular momentum is also metric dependent; all the other governing equations are topological.

The theory of mixed Hodge modules is applied to obtain results about the mixed Hodge structure on the intersection cohomology of a link of a subvariety in a complex algebraic variety. The main result, whose proof uses the purity of the intersection complex in terms of mixed Hodge modules, is a generalization of the semipurity theorem obtained by Gabber in the l-adic case. An application is made to the local topology of complex varieties.

Two aspects of the presentation of electromagnetism in three-dimensional space can be compared if a distinction is made between polar vectors P and axial vectors T. In option /spl alpha/, Baldomir and Hammond take B, E, D/spl alpha/, j/spl alpha/, H/spl alpha/ as basic vectors and need to use the *Hodge dual operator. Option /spl beta/ is presented based on the vectors B, E, D/spl beta/, j/spl beta/, H/spl beta/ where no special operator is required. For the presentation of electromagnetism in four-dimensional space the components of the group E B, common to both options, are brought together to form a tensor of 4/sup 2/ = 16 components. The B/sub ij/ are arranged within a 3 /spl times/ 3 square, with the E/sub k/ on the edge of this square. The slightly different forms of this arrangement correspond to the tensors identified symbolically by F/sub pq/ (B, E) for option /spl alpha/ and F/sub km/ (B, E) for option /spl beta/. Two Maxwell equations are related to the corresponding tensors for each option. For option /spl beta/, a tensor G/sup km/ directly linked to F/sub km/ (B, E) naturally leads to H/spl beta/ and D/spl beta/. The consideration of this tensor along with the four-dimensional current density vector J/sub /spl beta// (expressed using j/spl beta/ and /spl rho/) allows one to establish the two other Maxwell equations.

In this paper we propose a symplectic analogue of Barannikov's construction of semi-infinite Hodge structure variations over extended moduli spaces of complex structures. It is shown that the duality transformation for mirror torus fibrations over the same Monge-Ampere manifold isomorphically exchanges symplectic semi-infinite A-variations of Hodge structure introduced in this paper with Barannikov's semi-infinite B-variations of Hodge structure.

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariants

The asymptotics of integrals which depend on a critical point of a holomorphic function and the mixed Hodge structure in the vanishing cohomology are compared. Bibliography: 38 titles.

We study the Hodge spectral sequence of a local system on a compact, complex torus by means of the theory of harmonic integrals. It is shown that, in some cases, Baker's theorems concerning linear forms in the logarithms of algebraic numbers may be applied to obtain vanishing theorems in cohomology. This is applied to the study of Betti and Hodge numbers of compact analytic threefolds which are analogues of hyperelliptic surfaces. Among other things, it is shown that, in contrast to the two-dimensional case, some of these varieties are nonalgebraic. 0. Introduction. (0.0) The purpose of this paper is to determine the Hodge spectral sequence associated with a vector bundle with integrable connection on a complex torus. Such vector bundles typically arise as the hypercohomology sheaves attached to a proper and smooth morphism f: Y -* X where X is a complex torus. Consequently our results will have application to the study of those manifolds which admit such fiberings over tori. (0.1) In outline, the main result is as follows: Let (l, V) be a holomorphic vector bundle with integrable connection on a complex torus X = V/L, where L C V is a lattice in a complex vector space. Let W = QlV be the corresponding C-local system. We compute Hq(X, S2P(qll)) and Hk(X, W) in terms of harmonic differential forms via Hodge's theorem. First we show that qtS admits a global real analytic frame. Relative to this frame, C? differential forms in qIS may be represented by C?r m-vectors of differential forms p on V which are L-periodic (m = rank qLS'). Using the frame, a Hermitian metric may be introduced into the fibers of qLS in such a way that the harmonic theory takes on a simple form. Specifically, the Laplace equations lA 0 become, upon Fourier transform, AX(m).j4m) 0, m E Z2n (n = dimV). Each of these is a finite-dimensional matrix equation. We prove (for the choices made): (a) For Hk(X W), LA(m) j(m) 0 O has only the zero solution if m #7 0. Hence Hk(X, W) ker A(0). (b) For Hq(X, iP(qLS')), only a finite number of m can occur for which det i\(m) = 0 and these are identified by a diophantine condition involving the logarithms of the periods of X. These m are "singular". Therefore, modulo determining the singular m, all computations are reduced to a finite amount of linear algebra. Received by the editors October 28, 1980. 1980 Mathematics Subject Classification. Primary 14C30; Secondary 32J25, 14K20. 'Partially supported by NSF Grant MCS-800223 1. ?D 982 American Mathematical Society 0002-9947/82/0000-1051 /$04.75

An example is given, in dimension six, of a curvature tensor having positive sectional curvatures and negative Gauss-Bonnet integrand. A large class of questions in differential geometry involves the relationship between the topology and the geometry of a compact Riemannian manifold. One of these is the Hopf conjecture: If, in even dimensions, the sectional curvatures of such a manifold are positive, then so is the Euler number. The Hopf conjecture is known to be true in dimensions two and four by the following argument (Milnor, unpublished; [2]). One first writes down the Gauss-Bonnet formula, which, in every even dimension, equates the Euler number of the manifold to a certain integral over the manifold, where the integrand involves only the curvature tensor, and that only algebraically. One then shows (in dimensions two and four) that, at each point, positivity of the sectional curvatures implies positivity of this integrand. Most attempts to prove the full Hopf conjecture have been attempts to generalize this argument [1], [3], [4], [5], [6]. Thus, there arises the following, purely algebraic, question: Over a vector space of any even dimension, does a tensor having the symmetries of a curvature tensor and having positive sectional curvatures necessarily have positive Gauss-Bonnet integrand? We here answer this question in the negative. Fix a real, six-dimensional vector space V. A wedge denotes the wedge product, and a star a Hodge star operator.2 Denote by V2 the vector space of second-rank, antisymmetric tensors over V, by V2 its dual (the space of 2-forms over V), and by V22 the vector space of symmetric linear mappings from V2 to V2. We shall make use of the following fact: For any element A of V2, (1) ((A A A)* A (A A A)*)* 9(A A A A A)*A. For A any element of V2, denote by TA the element of V22 with action Received by the editors September 3, 1974. AMS (MOS) subject classifications (1970). Primary 53B20. i Supported in part by the National Science Foundation under contract GP-34721Xi, and by the Sloan Foundation. 2 Our conventions for the star operation are these: For any form A, A* = A; for B a 2-form and C a 4-form, B(C*) = C(B*) = (B A C)* = (B* A C*)*. Note that we introduce no metric on V. ? American Mathematical Society 1976 267 This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:16:18 UTC All use subject to http://about.jstor.org/terms

The Poincaré map of mixed exterior algebra generalizes the Hodge star operator and it plays a central rôle in the proofs of many classical identities of linear algebra. The principal purpose of this paper is to derive a new formula for it. This formula is useful in circumstances when the definition is too implicit. Several applications are discussed.

The authors compute explicitly all the Hodge numbers for all Calabi-Yau manifolds realised as complete intersections of hypersurfaces in products of complex projective spaces. This determines the essential part of the matter superfield spectrum for the heterotic superstring compactified on any of these manifolds. They use various techniques presented in the recent literature to obtain 265 distinct Hodge diamonds; they exemplify these techniques, giving all necessary details of their computations.